On generalized principal eigenvalues of nonlocal operators with a drift *

Abstract

This article is concerned with the following spectral problem: to find a positive function $\Phi$ $\in$ C 1 ($\Omega$) and $\lambda$ $\in$ R such that q(x)$\Phi$ (x) + ^ $\Omega$ J(x, y)$\Phi$(y) dy + a(x)$\Phi$(x) + $\lambda$$\Phi$(x) = 0 for x $\in$ $\Omega$, where $\Omega$ $\subset$ R is a non-empty domain (open interval), possibly unbounded, J is a positive continuous kernel, and a and q are continuous coefficients. Such a spectral problem naturally arises in the study of nonlocal population dynamics models defined in a space-time varying environment encoding the influence of a climate change through a spatial shift of the coefficient. In such models, working directly in a moving frame that matches the spatial shift leads to consider a problem where the dispersal of the population is modeled by a nonlocal operator with a drift term. Assuming that the drift q is a positive function, for rather general assumptions on J and a, we prove the existence of a principal eigenpair ($\lambda$ p , $\Phi$ p) and derive some of its main properties. In particular, we prove that $\lambda$ p ($\Omega$) = lim R$\rightarrow$+$\infty$ $\lambda$ p ($\Omega$ R), where $\Omega$ R = $\Omega$ $\cap$ (--R, R) and $\lambda$ p ($\Omega$ R) corresponds to the principal eigenvalue of the truncation operator defined in $\Omega$ R. The proofs especially rely on the derivation of a new Harnack type inequality for positive solutions of such problems.